Seminars and Colloquia by Series

Joint GT-UGA Seminar at UGA - A spectral sequence from Khovanov homology to knot Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 25, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Nathan Dowlin – Dartmouth
Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Using 2-torsion to obstruct topological isotopy

Series
Geometry Topology Seminar
Time
Monday, March 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Hannah Schwartz – Bryn Mawr
It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.

The nu+ equivalence class of genus one knots

Series
Geometry Topology Seminar
Time
Monday, March 4, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Kouki Sato – University of Tokyo
The nu+ equivalence is an equivalence relation on the knot concordance group. It is known that the equivalence can be seen as a certain stable equivalence on knot Floer complexes, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this talk, we show that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.

Joint GT-UGA Seminar at GT - Knots in homology spheres, concordance, and crossing changes

Series
Geometry Topology Seminar
Time
Monday, February 25, 2019 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris Davis – U Wisconsin Eau Claire
Any knot in $S^3$ may be reduced to a slice knot by making some crossing changes. Indeed, this slice knot can be taken to be the unknot. We show that the same is true of knots in homology spheres, at least topologically. Something more complicated is true smoothly, as not every homology sphere bounds a smooth simply connected homology ball. We prove that a knot in a homology sphere is null-homotopic in a homology ball if and only if that knot can be reduced to the unknot by a sequence of concordances and crossing changes. We will show that there exist knot in a homology sphere which cannot be reduced to the unknot by any such sequence. As a consequence, there are knots in homology spheres which are not concordant to those examples produced by Levine in 2016 and Hom-Lidman-Levine in 2018.

Joint GT-UGA Seminar at GT - Knot Traces and the Slice Genus

Series
Geometry Topology Seminar
Time
Monday, February 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lisa Piccirillo – UT Austin
Smooth simply connected 4-manifolds can admit second homology classes not representable by smoothly embedded spheres; knot traces are the prototypical example of 4-manifolds with such classes. I will show that there are knot traces where the minimal genus smooth surface generating second homology is not of the canonical type, resolving question 1.41 on the Kirby problem list. I will also use the same tools to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Heegaard Floer and the homology cobordism group

Series
Geometry Topology Seminar
Time
Monday, February 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jen Hom – Georgia Tech
We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

Simplification of singularities of Lagrangian and Legendrian fronts

Series
Geometry Topology Seminar
Time
Monday, February 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Álvarez-Gavela – IAS
We will present an h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. The h-principle says that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy. We will also discuss applications of the h-principle to symplectic and contact topology.

Acylindrical hyperbolicity of non-elementary convergence groups

Series
Geometry Topology Seminar
Time
Friday, February 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bin Sun – Vanderbilt
The notion of an acylindrically hyperbolic group was introduced by Osin as a generalization of non-elementary hyperbolic and relative hyperbolic groups. Ex- amples of acylindrically hyperbolic groups can be found in mapping class groups, outer automorphism groups of free groups, 3-manifold groups, etc. Interesting properties of acylindrically hyperbolic groups can be proved by applying techniques such as Monod-Shalom rigidity theory, group theoretic Dehn filling, and small cancellation theory. We have recently shown that non-elementary convergence groups are acylindrically hyperbolic. This result opens the door for applications of the theory of acylindrically hyperbolic groups to non-elementary convergence groups. In addition, we recovered a result of Yang which says a finitely generated group whose Floyd boundary has at least 3 points is acylindrically hyperbolic.

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